Wavelet Transform with GravMagPro

Wavelet Transform and Its Application in Interpreting Aeromagnetic Data

Introduction to Wavelet Transform

The wavelet transform is a powerful mathematical technique used for analyzing signals and datasets, particularly for examining localized variations and patterns. Unlike traditional Fourier transforms, which decompose a signal into sinusoids of varying frequencies, wavelet transforms break down a signal into wavelets—localized oscillations that can capture both frequency and location information. This property makes wavelet transforms particularly suitable for analyzing non-stationary signals, where frequency characteristics change over time.

Key Concepts

• Wavelets: Short, oscillatory functions that can be stretched or compressed. They provide a way to represent data with varying time-frequency resolutions.

• Continuous Wavelet Transform (CWT): A technique that transforms a signal into a two-dimensional representation, capturing information across various scales. The CWT is defined as:

  $ W(a, b) = \int_{-\infty}^{\infty} f(t) \psi^* \left( \frac{t - b}{a} \right) dt $

  where:

  • $( f(t) )$ is the input signal,
  • $( \psi )$ is the wavelet function,
  • $( a )$ is the scale,
  • $( b )$ is the translation.

• Scales: Refers to the degree of compression or stretching of the wavelet, allowing for analysis of different frequency components of the signal.

Applications of Wavelet Transform in Aeromagnetic Data Interpretation

Aeromagnetic data, which involves measuring variations in the Earth's magnetic field, is crucial in geophysical studies for identifying subsurface structures and mineral deposits. The application of wavelet transforms to aeromagnetic data offers several advantages:

1. Edge Detection

Wavelet transforms are particularly effective for edge detection in geophysical data. The ability to localize variations in magnetic intensity allows geophysicists to identify boundaries and discontinuities in geological formations. By analyzing wavelet coefficients, it becomes possible to detect features such as faults, fractures, and mineralized zones.

2. Noise Reduction

Aeromagnetic data often contains noise due to various environmental factors. The multi-resolution capability of wavelet transforms helps in separating signal from noise. By applying wavelet thresholding techniques, it is possible to denoise the data while retaining essential features, leading to clearer interpretations.

3. Upward Continuation

Upward continuation is a technique used to enhance the interpretation of aeromagnetic data by simulating the magnetic field at a higher altitude. This process helps in reducing the effects of noise and the influence of shallow sources, allowing for a clearer identification of deeper geological structures. The upward continuation enhances the signal of deeper anomalies, making it easier to distinguish between shallow and deep geological features.

4. Multi-Scale Analysis

The CWT provides a means to analyze data across multiple scales, enabling the identification of features that might not be visible at a single scale. For instance, large geological structures can be studied alongside smaller anomalies, giving a comprehensive view of subsurface conditions.

5. Highlighting Geological Structures

Wavelet transforms help in identifying various geological structures, including:

• Dykes: Vertical or near-vertical intrusions of magma that cut across existing strata. These can be detected as sharp edges in the wavelet coefficient plots.
• Sheets: Horizontal intrusions that spread over larger areas. They often appear as broader, flatter anomalies in the magnetic data.
• Points: Small, localized magnetic sources that can indicate features like mineral deposits. These are typically seen as peaks in the wavelet transform output.

6. Visualization

The output from wavelet transforms can be visualized as contour plots or surface maps, which represent the wavelet coefficients. This visual representation aids in interpreting complex data sets and enhances the understanding of spatial relationships in the magnetic field data.

Conclusion

The wavelet transform is a versatile and powerful tool for interpreting aeromagnetic data. Its ability to analyze localized variations, detect edges, and reduce noise makes it invaluable in geophysical exploration. By utilizing wavelet transforms, geoscientists can improve their interpretations of subsurface structures, leading to more informed decisions in resource exploration and geological mapping.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import GravMagPro as gmp

Load Corrected and RTP Aeromagnetic Data

df = pd.read_csv('mag_data11.csv')
df.head()

	X	Y	Z	mag
0	330000	6629200	211.055369	5535
1	330050	6629200	210.654999	5500
2	330100	6629200	210.286555	5476
3	330150	6629200	209.952679	5479
4	330200	6629200	209.655742	5496

plt.hist(df['mag'], bins = 50)

(array([4.000e+00, 6.000e+00, 2.000e+00, 2.000e+00, 9.000e+00, 1.200e+01,
        1.100e+02, 2.653e+03, 5.748e+03, 1.641e+03, 6.820e+02, 4.580e+02,
        3.590e+02, 2.940e+02, 2.710e+02, 1.760e+02, 1.380e+02, 6.500e+01,
        3.000e+01, 8.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00,
        0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00,
        0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00,
        0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00,
        0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00, 0.000e+00,
        0.000e+00, 2.000e+00]),
 array([3856.  , 3978.86, 4101.72, 4224.58, 4347.44, 4470.3 , 4593.16,
        4716.02, 4838.88, 4961.74, 5084.6 , 5207.46, 5330.32, 5453.18,
        5576.04, 5698.9 , 5821.76, 5944.62, 6067.48, 6190.34, 6313.2 ,
        6436.06, 6558.92, 6681.78, 6804.64, 6927.5 , 7050.36, 7173.22,
        7296.08, 7418.94, 7541.8 , 7664.66, 7787.52, 7910.38, 8033.24,
        8156.1 , 8278.96, 8401.82, 8524.68, 8647.54, 8770.4 , 8893.26,
        9016.12, 9138.98, 9261.84, 9384.7 , 9507.56, 9630.42, 9753.28,
        9876.14, 9999.  ]),
 <BarContainer object of 50 artists>)

df = df[df['mag'] < 7000]

grid_mag, grid_x, grid_y = gmp.make_grid(df, value_col = 'mag')

Original Aeromagnetic Data

gmp.plot_grid(grid_mag, grid_x, grid_y)

scales, coefficients, amplitudes, x_grid, y_grid = gmp.wavelet_transform(df, value_col = 'mag')

pad_size = 50

Downward contiuation using wavelet transform

gmp.plot_grid(amplitudes[1][pad_size:-pad_size, pad_size:-pad_size],x_grid, y_grid)

fig = gmp.plot_interactive_grid(amplitudes[1][pad_size:-pad_size, pad_size:-pad_size], x_grid, y_grid)
fig.show()

Upward continuation using wavelet transform

gmp.plot_grid(amplitudes[5][pad_size:-pad_size, pad_size:-pad_size],x_grid, y_grid)

fig = gmp.plot_interactive_grid(amplitudes[5][pad_size:-pad_size, pad_size:-pad_size], x_grid, y_grid)
fig.show()

Upward continuation to highlight deeper structures

gmp.plot_grid(amplitudes[15][pad_size:-pad_size, pad_size:-pad_size],x_grid, y_grid)

fig = gmp.plot_interactive_grid(amplitudes[15][pad_size:-pad_size, pad_size:-pad_size], x_grid, y_grid)
fig.show()